Line graph of a hypergraph

A hypergraph is linear if any two edges have at most one common vertex. Two edges are k-intersecting if they share at least k common vertices. A k-uniform hypergraph is hypegraph with with each edge of size k. Note that simple graphs are linear 2-uniform hypergraphs (a simple graph is loopless and contains no multiple edges). The intersection graph of a graph is usually called as Line Graph. The characterization of Line graphs was solved by Van Rooij and Wilf and by Beineke.